MAT 307 Numerical Analysis I
Computer arithmetic and floating point representations, conditioning, approximate solutions of nonlinear equations by bisection, Newton, secant, functional iteration methods, their rate of convergence and error analysis. Newton’s method applied to systems of nonlinear equations, Aitken acceleration process. Solving linear systems by LU factorization and Gaussian elimination, pivoting and scaling. Tridiagonal systems, matrix norms and matrix condition number, Solving linear systems by iterative methods Jacobi and Gauss Seidel iterative methods, matrix eigenvalue problems, power method, QR factrorization and SVD.
MAT 308 Numerical Analysis II
Vandermonde matrix, Lagrange interpolation, divided differences and finite difference interpolation methods, accuracy of interpolation, Chebyshev polynomials and optimum interpolation points, Hermite interpolation, Weierstrass approximation theorem, Bernstein polynomials, construction of interpolating splines, Cubic spline interpolation and B-splines. General least square problems and orthogonal polynomials. Taylor series, differentiation by interpolation and error analysis, Richardson extrapolation, numerical by trapezoid rule, Simphson rule and their error analysis, Gaussian quadrature rule.
MAT 309 Introduction to Harmonic Analysis
Fourier coefficients, summability in norm and homogeneous Banach spaces. Pointwise convergence, the order of magnitude of Fourier coefficients, Fourier series of square summable functions. Absolutely convergent Fourier series. Fourier coefficients of linear functionals. The convergence in norm. Convergence and divergence at a point. Set of divergence. Fourier transform in different spaces. Paley-Wiener theorems
MAT 310 Introduction to Number Theory
The famous problems of number theory, Diophantine equations. Pythagorean triples and the unit circle. Fermat’s last theorem. Divisibility and the greatest common divisor. Euclidean algorithm. Linear equations. Factorizations and the fundamental theorem of arithmetic. Congruences. Euler’s formula and Euler’s Phi function. Prime numbers, counting primes. Mersenne primes and perfect numbers. Powers modulo m and successive squaring (modular exponentiation). Computing kth roots modulo m. Powers, roots and unbreakable codes, RSA public key cryptosystem. Primality testing and Carmichael numbers. Sums of divisors. Powers modulo a prime p and primitive roots, indices. Squares modulo p, quadratic reciprocity law. Sums of two squares. The equation x4+ y4= z4. Square-triangular numbers. Pell’s equation x2+ Ny2=±1. Diophantine approximation. Continued fractions. The Gausssian integers and unique factorization. Irrational numbers and transcendental numbers. Multiplicative number theoretic functions, the Möbius inversion formula. p-adic numbers. Quadratic integers. The four square theorem. Algebraic integers. Quadratic fields and their integers. Norms and units of quadratic fields. Ideals in the ring of integers of quadratic fields and factorization of ideals into a product of prime ideals, ideal classes. Rational points on curves, cubic curves and elliptic curves. A rough sketch of the ideas in the proof of Fermat’s last theorem.
MAT 313 Elementary Algebraic Topology
Set theory and algebra, Analytic Topology, Elementary Homotopy Theory, Homotopy of Paths, Homotopy of Maps, Fundamental Group, Covering Spaces.
MAT 314 Measure theory and Lebesque integral
Systems of Sets. General Measure Theory. Measure on The Real Line. Measureble Functions. Convergence Almost Everywhere. Convergence in Measure. Egorov’s theorem. The Lebesgue Integral. Passage to the Limit in Lebesgue Integral. Definition and basic properties of L and L2 spaces.
MAT 316 Introduction to Approximation Theory
Lagrange interpolation, divided differences, accuracy of interpolation, Chebyshev polynomials and optimum interpolation points, divergence of equidistant points, Bivariate interpolation. Minimax approximation, Rational and Pade approximation, Peano’s theorem. Uniform approximation, Bohman-Korovkin Theorem, Bernstein polynomials, simultaneous approximation, modulus of continuity. B-splines, derivatives, Marsden identity. L2 approximations, orthogonal polynomials and convergence of Fourier series.
MAT 318 Mathematical Statistics
Probability, random variables and their distributions, special probability distributions, joint distributions, properties of random variables, functions of random variables.
MAT 320 Linear and Nonlinear Programming
Operations research, optimization problems, examples of linear programming problems, Mathematical background - linear algebra, Mathematical background – polyhedra, The simplex method, Duality, Complexity of algorithms and polynomial linear programming algorithms, Applications of linear programming to network flow problems, Nonlinear programming - convex programming and quadratıc programming.
MAT 321 Theory of Probability
Combinatorical analysis, axioms of probability, conditional probability and independence, random variables, limit theorems, additional topics in probability such as Poisson process entropy.
MAT 323 Graph Theory
Graphs. Isomorphism of graphs. Paths and cysles. Connectivity. Eulerian graphs. Hamiltonian graphs. Trees. Counting trees. Planar graphs. Euler’s formula. Graphs on other surfaces. Dual graphs. Infinite graphs. Colouring graphs. Colouring vertices. Colouring maps. History of the four color theorem and proof of the five color theorem. Colouring edges. Chromatic polynomials. Digraphs. Eulerian digraphs and tournaments. Markov chains. Some applications of graphs and trees. Hall’s ‘marriage’ problem. Transversal theory. Menger’s theorem. Network flows. Matroids.
MAT 326 Special functions and Differential Equations
Gamma, Bessel and Neumann functions. Properties of Bessel and Neumann functions. Bessel’s differential equation, Sturm-Liouville problem for Bessel’s differential equation. Legendre polynomials and their properties. Sturm-Liouville problems for Legendre’s differential equation. Properties of eigenvalues and eigenfunctions. Associated Legendre polynomials and their properties. Associated Legendre differential equation. Sturm-Liouville problems for associated Legendre differential equation. Spherical functions and their properties. Applications of special functions for solving initial boundary value problems for partial differential equations.
MAT 333 Basic Group Theory
Groups, subgroups, groups of permutations, cyclic groups, cosets and theorem of lagrange, direct products and finitely generated abelian groups, homomorphisms and factor groups, isomorphism theorems, free abelian groups.
MAT 338 Modern Physics II
Electric Fields and Gauss’s Law, Electric Potential, Capacitance and Dielectrics, Current and Resistance, Direct Current Circuits, Magnetic Fields and Their Sources, Faraday’s Law and Inductance, Alternating Current Circuits, Electromagnetic waves, Introduction to Special Theory of Relativity, Black Body Radiation, Photoelectric Effect, Compton Effect, The Atomic Models, Electron, Proton and Neutrons.
MAT 339 Modern Physics I
Introduction, Motion in One Dimension, Motion in Two Dimensions, The Laws of Motion, Work and Energy, Linear Momentum and Collisions, Rotational Motion, Oscillatory Motion, Wave Motion
MAT 342 Advanced Differential Geometry
Unit Speed Curves, Frenet Formulas, Arbitrary Speed Curves, Monge Curves, Evalutes and Involutes, Theory of Surfaces, Patch Computation, First fundamental form, Second fundamental form, Mean curvature and Gaussian curvature, Asymptotic Lines, Derivation of Gauss-Bonnet Formula, Osculating Sphere.